The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.
The talks should be of general interest to researchers and students in the mathematical sciences and related fields. All are welcome to attend.
There will be approximately three talks in this series each quarter, on Thursday afternoons at 4:00pm in Guggenheim 220(the Auditorium). Each talk will be followed by a reception.
Spring quarter, 2009
April 9, 2009:
Marty Golubitsky, Ohio State University
Symmetry-Breaking; Synchrony Breaking
Abstract:
A coupled cell system is a network of interacting dynamical Coupled cell models assume that the
output from each cell is important and that signals from two or more cells can be compared so that
patterns of synchrony can We ask: which part of the qualitative dynamics observed in coupled cells
is the product of network architecture and which part depends on the specific equations? In our theory,
local network symmetries replace symmetry as a way of organizing network dynamics, and synchrony-breaking
replaces symmetry-breaking as a basic way in which transitions to complicated dynamics occur. Background on symmetry-breaking and pattern formation will be presented.
Marty Golubitsky is Distinguished Professor of Mathematics and Physical Sciences at the Ohio State University, where he serves as Director of the Mathematical Biosciences Institute. He received his PhD in Mathematics from M.I.T. in 1970 and has been Professor of Mathematics at Arizona State University (1979-83) and Cullen Distinguished Professor of Mathematics at the University of Houston (1983-2008).Dr. Golubitsky works in the fields of nonlinear dynamics and bifurcation theory studying the role of symmetry in the formation of patterns in physical systems and the role of network architecture in the dynamics of coupled systems. His recent research focuses on some mathematical aspects of biological applications: animal gaits, the visual cortex, the auditory system, and coupled systems. He has co-authored four graduate texts, one undergraduate text, and two nontechnical trade books, (Fearful Symmetry: Is God a Geometer with Ian Stewart and Symmetry in Chaos with Michael Field) and over 100 research papers. Dr. Golubitsky is a Fellow of the American Academy of Arts and Sciences, a Fellow of the American Association for the Advancement of Science (AAAS), the 1997 recipient of the University of Houston Esther Farfel Award, and the 2001 co-recipient of the Ferran Sunyer i Balaguer Prize (for The Symmetry Perspective). He has been elected to the Councils of the Society for Industrial and Applied Mathematics (SIAM), AAAS, and the American Mathematical Society. Dr. Golubitsky was the founding Editor-in-Chief of the SIAM Journal on Applied Dynamical Systems and has served as President of SIAM (2005-06).
May 21, 2009:
Nick Trefethen, Oxford University
CHEBFUNS: A NEW KIND OF NUMERICAL COMPUTING
Abstract:
Chebfuns represent a new kind of computing
that aims to combine the feel of symbolics with the speed of precision
numerics. The idea is
to represent functions by piecewise Chebyshev expansions whose
length is determined adaptively to maintain an accuracy of close to
machine. The software is implemented in object-oriented Matlab, with
familiar vector operations such as sum and diff overloaded
to analogues for functions such as integration and
differentiation, and the chebop extension solves linear ordinary differential
equations by typing a backslash. This is joint work with others
including Zachary Battles, Folkmar Bornemann, Toby Driscoll, Ricardo
Pachon, and Rodrigo Platte.
Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group at Oxford University. A Fellow of the Royal Society and a member of the National Academy of Engineering, he is known for books, articles, and software in areas including numerical linear algebra, transition to turbulence, approximation of functions, numerical conformal mapping, and spectral methods for partial differential equations.
Winter quarter, 2009
January 29, 2009:
Claude Le Bris, Paris
Mathematical challenges in Molecular simulation: an overview
Abstract:
Molecular simulation is increasingly important in many engineering sciences
and life sciences. The field has only been recently explored by
mathematical analysts and numerical analysts, leading to several
achievements, but also leaving major challenging issues unsolved, both
theoretically and computationally.
The talk will present the state of the art and will review major
mathematical issues of practical importance and theoretical relevance. It
will also relate such issues of molecular simulation with issues in
materials science. It is mostly based on a recent article coauthored with
E. Cances and PL. Lions, and published in Nonlinearity, volume 21,
T165-T176, 2008.
Claude Le Bris is a Professor of Applied Mathematics at the Ecole Nationale des Ponts et Cahussees, Paris. He is also Civil engineer-in-chief, Associate Professor at the Ecole Polytechnique and scientific director of the MICMAC project (multiscale methods) at INRIA. Professor Le Bris has won numerous awards including the Blaise Pascal Prize 1999 from the French Academy of Sciences, the CS 2002 Prize in Scientific computing from Communications & Systems, and the Giovanni Sacchi-Landriani Prize 2002 from the Lombard Academy of Arts and Sciences.
February 12, 2009:
Thomas Hou, Caltech
Recent Progress on Dynamic Stability and Global Regularity
of 3D Incompressible Euler and Navier-Stokes Equations
Abstract:
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial
data is one of the seven Millennium Open Problems posted by the Clay Mathematical Institute. We review some
recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle
dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The
local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex
stretching, thus preventing a finite time singularity. Our studies also reveal a surprising stabilizing effect of convection for the 3D incompressible Euler and Navier-Stokes equations. Finally, we present a new class of solutions for the 3D
Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property by exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear
stability and the global regularity of this class of solutions.
Thomas Hou is the Charles Lee Powell Professor and Executive Officer of Applied and Computational Mathematics at the California Institute of Technology. Professor Hou has won numerous awards including the Sloan Fellowship (1990-1992), the Feng Kang Prize in Scientific Computing (1997), the APS Francois N. Frenkiel Award (1998), the SIAM James H. Wilkinson Prize in Numerical Analysis and Scientific Computing (2001), the Morningside Gold Medal in Applied Mathematics, International Congress of Chinese Mathematicians (2004), and the Computational and Applied Sciences Award, the United States Association of Computational Mechanics (2005). In addition, he is on numerous editorial boards including as the founding editor of the SIAM Journal on Multiscale Modeling and Simulation.
February 26, 2009:
Carl Bender, Washington University
Quantum Mechanics in the Complex Domain
Abstract:
The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H=p^2+ix^3,for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian H=p^2+ix^3 is
not Dirac Hermitian, it is PT symmetric; that is, it is symmetric under combined space reflection P and time reversal T.
In general, if a Hamiltonian H is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for determining the adjoint operation under which H is Hermitian.
It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric g^\mu\nu in curved space is before solving Einstein's equations.) In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was
the appearance of ghost states (states of negative norm). The cause of the disease is that the Hamiltonians for these models were inappropriately treated as if they were Dirac Hermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories.
Carl Bender is a Professor of Physics at Washington University in St. Louis. Professor Bender has won numerous awards including the Sloan Fellowship (1972-1977), the M.I.T. Graduate Student Council Teaching Award (1976), a Fulbright Fellowship to the United Kingdom (1995-1996), John Simon Guggenheim Memorial Foundation Fellowship (2003-2004), the Ulam Fellowship, Los Alamos National Laboratory, (2006-2007), the Compton Faculty Achievement Award, Washington University (2007), and Wilfred R. and Ann Lee Konneker Distinguished Professor of Physics (2007). In addition, he is a co-author of one of the most influential texts in applied mathematics: Advanced Mathematical Methods for Scientists and Engineers (Bender & Orzag).
Autumn quarter, 2008
October 9, 2008:
David Keyes, Columbia University,
Attacking the Asymptotic Algorithmic Bottleneck:
Scalable Solvers for Scientific Simulation
Abstract:
Many simulations with promise for scientific
discovery and technological advance are of multiscale and multiphysics
character, and subject to uncertainties of model or data. Asymptotically in
mesh resolution, such applications are ultimately bottlenecked by solution
algorithms. Simulations based on Eulerian formulations of partial
differential equations can be among the first applications to take advantage
of petascale capabilities, but not the way most are presently pursued.
Weak
scaling avoids the fundamental limitation expressed in Amdahl's Law but only
optimal implicit formulations can get around another limitation on scaling
that is an immediate consequence of Courant-Friedrichs-Lewy stability theory
under weak scaling of a PDE. Many PDE-based applications and other
lattice-based applications with petascale roadmaps (such as quantum
chromodynamics) will likely be forced to adopt optimal implicit solvers.
However, even this narrow path to petascale simulation is made treacherous
by the imperative of dynamic adaptivity, which drives us to consider
algorithms that are less synchronous than those in common use today. After
reviewing themes in scalable algorithms at a high level, we focus on some
particular advances in the U.S. magnetic fusion energy program and other
PDE-based applications enabled by scalable solution algorithms.
David E. Keyes is the Fu Foundation Professor of Applied Mathematics in the Department of Applied Physics and Applied Mathematics at Columbia University, and the Chair-designate of the Mathematical and Computer Sciences and Engineering Division at KAUST. With backgrounds in engineering, applied mathematics, and computer science, Keyes works at the algorithmic interface between parallel computing and the numerical analysis of partial differential equations, across a variety of applications. Newton-Krylov-Schwarz parallel implicit methods, introduced in a 1993 paper, are now widely used throughout computational physics and engineering and scale to many thousands of processors. Keyes is currently the Vice President-at-Large of SIAM, and a member of the advisory committees of the Mathematics and Physical Sciences Directorate and the Office of CyberInfrastructure of the NSF.
October 23, 2008:
John Bush, MIT
Interfacial Biomechanics
Abstract:
We examine several biological systems dominated by the
influence of surface tension. Particular attention is given to elucidating
natural strategies for
water-repellency, underwater breathing, fluid transport on a small scale,
and walking on water.
Examples are primarily taken from the world of insects, but capillary
feeding in shorebirds is also
highlighted, and the mechanics of spider capture silk touched upon. A number
of Nature's designs
are rationalized and serve as inspiration for biomimetic microfluidic
devices.
John Bush is on the faculty in the Mathematics Department at MIT and is Director of the department's Fluid Dynamics Laboratory. His research involves an interplay between experimental and theoretical modeling techniques, and is focussed towards identifying and elucidating new fluid dynamical phenomena. He is particularly interested in Surface Tension-Driven Phenomena and Biofluidynamics. Among his distinctions, Professor Bush received the Gallery of Fluid Motion Award of the APS Division of Fluid Mechanics virtually every year since 1999. He received an NSF Career Award in 2002. In 2003, the department faculty selected him to be the initial holder of the Edward F. Kelly Research Award.
November 6, 2008:
Mark Lewis, University of Alberta
Dynamics of emerging wildlife diseases
Abstract: In this talk I will present recent progress in modelling the dynamics of emerging wildlife diseases. I will focus on two examples, one involving interactions between hosts (birds) and disease vectors (mosquitoes) in the outbreak of West Nile virus, and the other involving a "spill over" and "spill back" disease between net pen aquaculture and wild salmon. The focus of the talk will be quantitative assessment of the disease dynamics using dynamical systems, and the resulting interplay between models and data.
Mark
Lewis
holds the Canada Research Chair in Mathematical Biology at the
University of Alberta. Dr. Lewis' research is in mathematical biology and
ecology, including modelling and analysis of nonlinear PDE and integral
models in population dynamics and ecology. Applications, made to case
studies with detailed data and biology, include: wolf territories, spatial
spread and impact of introduced pest species, wildlife diseases, vegetation
shift in response to climate change and recolonization of Mount St. Helens.
Dr. Lewis obtained his doctorate from the University of Oxford in 1990 in Mathematical Biology. He was a faculty member at the University of Utah until 2001, and has also held visiting and research fellowships at Princeton University and Imperial College, University of London. He is Past President of the Society for Mathematical Biology. He is Chief Editor of the Journal of Mathematical Biology, and serves on editorial boards of six other journals. His research has been recognized by a Sloan Research Fellowship, a National Young Investigator Award (US NSF), Killam and McCalla Professorships, the American Society of Naturalists Presidential Award and the Lee Segel Prize.